Question

A stationary wave is represented by $y = 10\sin\left(\dfrac{\pi x}{4}\right)\cos(20\pi t)$, where $x$ and $y$ are expressed in cm and $t$ in second. Distance between two consecutive nodes is

• A. $4\,\text{cm}$
• B. $1\,\text{cm}$
• C. $8\,\text{cm}$
• D. $2\,\text{cm}$

Hint:

In a stationary wave, nodes are separated by half the wavelength.

Answer 1

The Correct Option is A

Step 1: Identify wave number.
From the given equation:
\[ y = 10\sin\left(\frac{\pi x}{4}\right)\cos(20\pi t) \] The wave number is:
\[ k = \frac{\pi}{4} \]

Step 2: Relation between wavelength and wave number.
\[ k = \frac{2\pi}{\lambda} \Rightarrow \lambda = 8\,\text{cm} \]

Step 3: Distance between consecutive nodes.
Distance between two consecutive nodes is $\dfrac{\lambda}{2}$.
\[ \frac{8}{2} = 4\,\text{cm} \]

Step 4: Conclusion.
The distance between two consecutive nodes is $4\,\text{cm}$.